Isometries of combinatorial Tsirelson spaces

2022/08/31 by Maślany, Natalia
#46B04 #46B25 #46B45 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2209.00113

Abstract

We extend existing results that characterize isometries on the Tsirelson-type spaces T[(1)/(n), S₁] (n∈ ℕ, n≥ 2) to the class T[θ, S_α] (θ∈ (0, (1)/(2)], 1\leqslant α< ω₁), where S_α denote the Schreier families of order α. We prove that every isometry on T[θ, S₁] (θ∈ (0, (1)/(2)]) is determined by a permutation of the first \lceil θ^-1 \rceil elements of the canonical unit basis followed by a possible sign-change of the corresponding coordinates together with a sign-change of the remaining coordinates. Moreover, we show that for the spaces T[θ, S_α] (θ∈ (0, (1)/(2)], 2\leqslant α< ω₁) the isometries exhibit a more rigid character, namely, they are all implemented by a sign-change operation of the vector coordinates.

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